Opposite category

Examples

• An example comes from reversing the direction of inequalities in a partial order. So if X is a set and ≤ a partial order relation, we can define a new partial order relation ≤^op by
• Given a semigroup, one usually defines the opposite semigroup as ^op = where x*y ≔ y·x for all x,y in S. So also for semigroups there is a strong duality principle. Clearly, the same construction works for groups, as well, and is known in ring theory, too, where it is applied to the multiplicative semigroup of the ring to give the opposite ring. Again this process can be described by completing a semigroup to a monoid, taking the corresponding opposite category, and then possibly removing the unit from that monoid.
• The category of Boolean algebras and Boolean homomorphisms is equivalent to the opposite of the category of Stone spaces and continuous functions.
• The category of affine schemes is equivalent to the opposite of the category of commutative rings.
• The Pontryagin duality restricts to an equivalence between the category of compact Hausdorff abelian topological groups and the opposite of the category of abelian groups.
• By the Gelfand–Neumark theorem, the category of localizable measurable spaces is equivalent to the category of commutative Von Neumann algebras.

  Properties